How do I plot a series of disks in the center of each hexagon?

Just reuse the expression for the x and y components that you used when centering the hexagons:

Table[
 Graphics[Disk[{3 i + 3 ((-1)^j + 1)/4, Sqrt[3]/2 j}, {0.25, 0.25}]],
 {i, 0, 1},
 {j, 0, 3}
 ]

This gives you:

enter image description here

Clear["Global`*"]

You can use CirclePoints to define the Polygon

h[x_, y_] := Polygon[{x, y} + # & /@ CirclePoints[6]]

EDIT: As pointed out in the comment by J.M., using RegularPolygon this can be simplified to

h[x_, y_] := RegularPolygon[{x, y}, 1, 6]

The common center points are

ctrs = Table[
    {3 i + 3 ((-1)^j + 1)/4, Sqrt[3]/2 j},
    {i, 0, 1}, {j, 0, 3}] //
   Flatten[#, 1] &;

Map Polygon and Disk onto the centers.

Graphics[{
  EdgeForm[Opacity[.7]], {
     White, h @@ #,
     Black, Disk[#, 0.25]} & /@
   ctrs}]

enter image description here

1. You can post-process your p1 to add disks at RegionCentroid of polygons:

p1 /. p_Polygon:> {p, FaceForm[Black], Disk[RegionCentroid @ p, {1, 1} / 4]}

enter image description here

2. Alternatively, you can create a single disk/hexagon pair centered at {0, 0}

prims = {Black, Scale[Disk[], {1, 1}/4], 
   EdgeForm[Gray], FaceForm[], RegularPolygon[{0, 0}, 1, 6]};

and Translate it using centers:

centers = {3 # + 3 (1 + (-1)^#2)/4, Sqrt[3] #2/2} & @@@ 
   Tuples[{{0, 1}, Range[0, 3]}];


Graphics[Translate[prims, #] & /@ centers]

enter image description here

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